Orbit Dirichlet series and multiset permutations

Orbit Dirichlet series and multiset permutations We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz’s q-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series’ abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monatshefte f�r Mathematik Springer Journals

Orbit Dirichlet series and multiset permutations

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Publisher
Springer Vienna
Copyright
Copyright © 2017 by Springer-Verlag GmbH Austria
Subject
Mathematics; Mathematics, general
ISSN
0026-9255
eISSN
1436-5081
D.O.I.
10.1007/s00605-017-1128-9
Publisher site
See Article on Publisher Site

Abstract

We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz’s q-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series’ abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture.

Journal

Monatshefte f�r MathematikSpringer Journals

Published: Nov 4, 2017

References

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